Chapter 28 Problem 34

Problem

A wire, in a plane, has two arcs of a circle connected by radial lengths of wire. Determine ${\vec {B}}$ at point $C$ in terms of $R_{1}$ , $R_{2}$ , $\theta$, and the current $I$

Solution

Since the point C is along the line of the two straight segments of the current, these segments do not contribute to the magnetic field at C.

${\vec {B}}={\frac {\mu _{0}I}{4\pi }}\int _{0}^{R_{1}\theta }{\frac {d{\vec {l}}\times {\hat {r}}}{R_{1}^{2}}}+{\frac {\mu _{0}I}{4\pi }}\int _{R_{2}\theta }^{0}{\frac {d{\vec {l}}\times {\hat {r}}}{R_{2}^{2}}}$ $={\frac {\mu _{0}I}{4\pi R_{1}^{2}}}{\hat {k}}\int _{0}^{R_{1}\theta }ds+{\frac {\mu _{0}I}{4\pi R_{2}^{2}}}{\hat {k}}\int _{R_{2}\theta }^{0}ds$ $={\frac {\mu _{0}I\theta }{4\pi R_{1}}}{\hat {k}}+{\frac {\mu _{0}I\theta }{4\pi R_{2}}}{\hat {k}}$

$={\frac {\mu _{0}I\theta }{4\pi }}\left({\frac {R_{2}-R_{1}}{R_{1}R_{2}}}\right){\hat {k}}$